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Appendix II.

We thus have ist, 2nd, and 3rd moments of an area, each of
which has its use, and all easily found by a graphic construction.
Referring to Fig. 809, let the given area PPP be imagined to

L2    LI      L,

rotate round axis xx. Draw PPI II xx, and take any new line,
xtXi at any distance h, and I! xx. Project PXM and PN, and join
M N, giving a point QJ. Do this for several horizontal intercepts,
and obtain the shaded 1st moment area PCh In like manner
project Q^ and join I^N, giving the shaded 2nd moment area PQ2.
Similarly the 3rd moment area is obtained from area PQ2. Then,
calling the areas respectively A! , A2 , and A3 ,

ist  moment = Ax h   \

2nd moment = A9 h? > of the original area round x x,

3rd moment = A3 A3 )

and any higher-powered moment can be obtained by continued

Proof.  Let p P! be an element of area having width <?.
nih moment = S{PP   x e x (PN)*}




and LM

and ist moment = 2{ppx x e x (PN)J}
i x PN x e)  h x S(<? x PQi) = A^.



= -            and LLt